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© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license.

Prof Steve EasterbrookDept of Computer Science

http://www.cs.toronto.edu/~sme/SystemsThinking

CSC2720H:Systems Thinking for

Global Problems

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 2

Week 8➜Group Presentations!

➜Review of Chaos and Complexity Theory

➜Self-Organized Criticality➜Population Models➜Self-organised criticality

➜Leverage Points

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 3

Complex Systems Theory - Key concepts➜Non-linear Dynamical Systems

ÄChange in inputs/parameters lead to disproportionate change in outputs

➜Sensitivity to Initial ConditionsÄThe “butterfly effect”ÄSmall differences in initial conditions tend to grow exponentially

➜Attractors (Simple and Strange)ÄChaotic systems are not randomÄCan predict the shape of the attractor, but not specific future states

➜Self-similarity and Fractals

➜Criticality and Tipping Points

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 4

Feigenbaum RatioNamed for mathematician Mitchell J. Feigenbaum

The Feigenbaum ratio is a bifurcation fractal produced by a period-doubling cascade, such as in the logistic bifurcation diagram.

The period doubling comes at a constant rate given by the delta constant: 4.66921166091029 etc.

This constant arises in any dynamical system that approaches chaotic behavior via period-doubling bifurcations:

fluid-flow turbulence, electronic oscillators, chemical reactions, the Mandelbrot set.

The Harder a System is Pushed theFaster the Change Comes

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 5

2.5 2.9 3.0 3.3

Logistic equations, plotted to show only X values from 20-25.

As a bifurcation approaches the system takes progressively longer to attenuate.

For r=3.3 the system attenuates to one value after 5 iterations.

For r = 3.0 the system is still attenuating at the decimal places 4 - 6 after 1 million iterations.

The bifurcation occurs because the instability grows until it flies apart.

Stability at1 value

IncreasingInstability

Vibrating so hardIt flies apart

Return tostability, but with

2 stable points

Tipping points and Instabilities

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 6

Formulated by George Kingsley Zipf who observed some striking patterns in human systems.

Examples:

Word usage in the English (and other) languages,

Distribution of city population sizes.

Zipf’s Law

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 7

Energy100 101 102 103 104 105 106 107 108 109

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107 A Power Law Distribution

Freq

uenc

y

Lots of small

events

Very rare large

events

Do very little work

Do the vast majority of the work

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 8

Contributors to Open Source Projects

Xu, J., et al. (2005). A Topological Analysis of the Open Source Software Development community. In Proceedings of the 38th Hawaii International Conference on System Sciences - 2005

Data from 160,000 Sourceforge projects

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 9

Power Law Relationships – Cotton Prices

Mandelbrot’s (1963) analysis of monthly variations in cotton pricesduring a 30 month period. The left plot shows the month by monthchanges. Note how they vary; lots of small changes, and fewer largechanges. The right logarithmic graph shows the same data is a power-law distribution, indicating the cotton commodities market is at thecritical level (SOC). Other commodities follow a similar pattern.

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 10

Power Law Relationships – Natural Disasters

http://pubs.usgs.gov/fs/natural-disasters/figures/fig3.html

Plot of cumulative frequency of dollar loss due to earthquakes and hurricanes in the U.S. between 1900 and 1989. Data presented in this manner reveal linear trends which provide the basis for forecasting the probability of future dollar loss.

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University of Toronto Department of Computer Science

© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license. 11

Self-Organized Criticality

Simple model of how avalanches work:Bak, Tang, and Wiesenfeld’s Sandpile model

https://www.youtube.com/watch?v=h737qbQRPME&feature=youtu.be&t=4m

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© 2020 Steve Easterbrook. This presentation is available free for non-commercial use with attribution under a creative commons license.

Game

Triangles

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